$(45$ points$)$ Given a single training sample $)$ for softmax regression $($multi$-$class

classification$)$ of three classes. Current parameters for the three classes are ${}_1=[5,3,-2{]}^T,{}_2=$

$[4,-2,1{]}^T,{}_3=[-5,2,3{]}^T.$ Note that we assume that we extend $x$ with a $0-$th dimension, i$.$e$.,x=$

$[1,x_1,x_2{]}^T,$ where $x_0$ is a constant $0.$

$($a$)(15$ points$)$ Find the cross$-$entropy cost function value for this training sample.

$($b$)(15$ points$)$ Find negative gradients of ${}_1,{}_2,{}_3,$ with this training sample and the cross$-$entropy

cost function.

$($c$)(15$ points$)$ If a learning rate $=0\mathrm{.}1$ is used, what would be parameters ${}_1,{}_2,{}_3$ after $1$ iteration

of gradient descent? Please note that all parameter dimensions are updated jointly in one iteration.

$($d$)(15$ points$)$ Find the cross$-$entropy cost function value for this training sample after $1$ iteration of

gradient descent in $($c$).$